Could you please help me with any information which would allow me to get an explicit solution for this equation?
It is an implicit solution to an optimization problem involving resource allocation in a game-theoretic setting. (I do not think this type of problems have been considered.)
$$y'(x)= 1-\frac{2\sqrt{2 y(x)}}{\int_{0}^{x} \frac{dt}{\sqrt{y(t)+y(x)}}} $$
Which class of integro-differential equations would it belong - so that I could search for a solution?
It is a specific parametric example for a more general problem $$(2 y(x))^a f(x) = (1-y'(x)) \int_{x_{min}}^{x} a (y(t)+y(x))^{a-1} d F(t)$$ (given $a$ and distribution $F(x), f(x)=F'(x)$).
For $a=1$, the explicit solution is strikingly elegant: $$y(x) = \int_{x_{min}}^x \left(\frac{F(t)}{F(x)}\right)^2 dt$$
Many thanks in advance